11/28/2020

NON-JUVENILE ASYMPTOTICS

  • Team member EDUARD GIMENEZ
  • Team member PATRICK ALTMEYER
  • Team member SIMON NEUMEYER
  • Team member JAKOB PÖRSCHMANN

Overview

  • Data exploration
  • Methodology
  • Forecasts
  • Interpretation of results

Data exploration

Time series (normalized)

Country clusters

Existing observations

Methodology

Framework

  1. Fit classifier.fit(X,y) in parallel for all \(m \in [1,24]\):

\[ \begin{aligned} &&v_m&=\text{model}(\mathbf{X}_m) \\ \text{where}&& \mathbf{X}_m &= \begin{pmatrix} \mathbf{C}_1 & v_{1,-137} & ... & v_{1,m-1} \\ ... & ... & ... & ... \\ \mathbf{C}_{n_m} & v_{n_m,-137} & ... & v_{n_m,m-1} \end{pmatrix} \end{aligned} \]

  1. Predict classifier.predict(X) recursively for all \(m \in [1,24]\) where

\[ \begin{aligned} && \mathbf{X}_m = \begin{pmatrix} \mathbf{C}_1 & v_{1,-137} & ... & \hat{v}_{1,m-1} \\ ... & ... & ... & ... \\ \mathbf{C}_{n_m} & v_{n_m,-137} & ... & \hat{v}_{n_m,m-1} \end{pmatrix} \end{aligned} \]

Confidence interval

  • Normalised predictions are (largely) \(\hat{\mathbf{v}}\in[0,1]\).
  • We aim for small bands near 1 and 0 and wider bands elsewhere.
  • Predictions can be thought of as being generated from a beta distribution with \(\mathbb{E}(\mathbf{v})=\frac{\alpha}{\alpha+\beta}\)
  • Fix \(\beta=5\) and compute \(\alpha=\frac{\mathbf{v}\beta}{(1-\mathbf{v})}\) and then
L = sc.betaincinv(alpha, beta, .075)
U = sc.betaincinv(alpha, beta, .925)

Beta distribution

Score function

PLACEHOLDER

Forecasts

PLACEHOLDER

Interpretation

PLACEHOLDER